In the 1920’s, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds

around the concept of infinity. Imagine a hotel with an infinite

number of rooms and a very hardworking night manager. One night, the Infinite Hotel

is completely full, totally booked up

with an infinite number of guests. A man walks into the hotel

and asks for a room. Rather than turn him down, the night manager decides

to make room for him. How? Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every guest moves from room number “n” to room number “n+1”. Since there are an infinite

number of rooms, there is a new room

for each existing guest. This leaves room 1 open

for the new customer. The process can be repeated for any finite number of new guests. If, say, a tour bus unloads

40 new people looking for rooms, then every existing guest just moves from room number “n” to room number “n+40”, thus, opening up the first 40 rooms. But now an infinitely large bus with a countably infinite

number of passengers pulls up to rent rooms. countably infinite is the key. Now, the infinite bus

of infinite passengers perplexes the night manager at first, but he realizes there’s a way to place each new person. He asks the guest in room 1

to move to room 2. He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on. Each current guest moves

from room number “n” to room number “2n” — filling up only the infinite

even-numbered rooms. By doing this, he has now emptied all of the infinitely many

odd-numbered rooms, which are then taken by the people

filing off the infinite bus. Everyone’s happy and the hotel’s business

is booming more than ever. Well, actually, it is booming

exactly the same amount as ever, banking an infinite number

of dollars a night. Word spreads about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an infinite line

of infinitely large buses, each with a countably infinite

number of passengers. What can he do? If he cannot find rooms for them,

the hotel will lose out on an infinite amount of money, and he will surely lose his job. Luckily, he remembers

that around the year 300 B.C.E., Euclid proved that there

is an infinite quantity of prime numbers. So, to accomplish this

seemingly impossible task of finding infinite beds

for infinite buses of infinite weary travelers, the night manager assigns

every current guest to the first prime number, 2, raised to the power

of their current room number. So, the current occupant of room number 7 goes to room number 2^7, which is room 128. The night manager then takes the people

on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seat

number on the bus. So, the person in seat

number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers. All the buses’ passengers

fan out into rooms using unique room-assignment schemes based on unique prime numbers. In this way, the night

manager can accommodate every passenger on every bus. Although, there will be

many rooms that go unfilled, like room 6, since 6 is not a power

of any prime number. Luckily, his bosses

weren’t very good in math, so his job is safe. The night manager’s strategies

are only possible because while the Infinite Hotel

is certainly a logistical nightmare, it only deals with the lowest

level of infinity, mainly, the countable infinity

of the natural numbers, 1, 2, 3, 4, and so on. Georg Cantor called this level

of infinity aleph-zero. We use natural numbers

for the room numbers as well as the seat numbers on the buses. If we were dealing

with higher orders of infinity, such as that of the real numbers, these structured strategies

would no longer be possible as we have no way

to systematically include every number. The Real Number Infinite Hotel has negative number rooms in the basement, fractional rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1. Square root rooms, like room radical 2, and room pi, where the guests expect free dessert. What self-respecting night manager

would ever want to work there even for an infinite salary? But over at Hilbert’s Infinite Hotel, where there’s never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night manager serve to remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity. Maybe you can help tackle these problems after a good night’s sleep. But honestly, we might need you to change rooms at 2 a.m.

Que??????????

Why don’t the new people just go to the next upon Tom on a higher floor, people would give the hotel bad ratings if they had to move out all the time

Why can't you just add guests at the end? Because there is no end to the number of rooms, right? But then how do you ask anyone to change rooms? How does that open up space by moving existing guests to space which we've already established doesn't exist?

سحب طاقة البطاريه ….

السيخ بتاع رقلئ مالوٌا ….

For anyone interested in the applications of this paradox, it’s hugely relevant to set theory. It is for this exact reason that their is a distinction between countable and uncountable infinity. It’s also a major part of the use for the axiom of choice which is one of the most basic (basic meaning foundational not necessarily simple) rules that defines set theory

That's why infinity to exist, there should be only 1 infinity where the rest are things.

Poor cleaning lady

Or…

Since there’s an infinite number of rooms, instead of asking every single person to move, just move the new person to the infinity-eth room!

Also how can there be an infinite number of people at the hotel?

And there has to be other people in the world!

And what the heck is infinity times infinity?

Infinite infinity?!

How does infinity work?!

How does anything work??!!

WHAT IS LIFE?!Why bother making more room if you already have an infinite income?

if i had a hotel room for every gender

Don't worry i'll just grab my stuff real quick at room 11.6 septillion

David David David,

You're trying to hard.

If the night manager keeps changing the rooms infinite number of guests won’t come back

はぇ〜すんごい

This riddle simply cannot exist even from the most illogical point of view.

INSTEAD OF MOVING ROOM1 TO ROOM2,HE MUST PUT THE CUSTOMER TO THE NEXT NUMBER LIKE "ROOM100 IS OCCUPIED SO GO TO ROOM 101" AND SO ON..HE JUST MAKE THINGS COMPLICATEDthis hotel wouldve won guiness world records such as having the first building to reach space, the most rooms in a hotel and the longest elevator ride in a hotel

What if someone booked hotel and he have to reach his room. So he must have to travel to infinity. Something wrong..

Ok but honestly this is the stupidest paradox ever.

The infinite hotel if full

Also ted ed SO WE'RE GONNA MOVE EVERYONE TO ANOTHER ROOM BECAUSE WE CAN

This night manager must have a godlike powers to manage the infinite hotel and the first bus with infinite number of costumer simultaneously in a single night

2:53 Saggy boobs

moving from room to room can cause infinite customers angry which could result in infinity war.

I learned this stuff in my Discete math class

Wow

Manager:You need to take the stairs to reach floor 82758282838385929295932944

How are there infinite people? Also Earth is round how is there an infinite bus without crashing into itself?

Wait but if the infinite hotel is full, that means there is no more rooms. How could the so called "last person" move to the next room when it is already full?

They jst freakn complicated it with numbers. I get the idea of infinite.. they could have just put the new guest on top. (-_-')

Just say :

Hotel room was overbooked! Please get another hotel Sir!Case ClosedAn infinite hotel could never be full in the first place.

Hotel? TrivaGO TO ANOTHER ROOM

Dude, this paradoxes never fail at disappointing

5:42

Can someone explain that?

this is so much confusing

seriously who invented math?

Nonsense. What's the point? Want to be a God? Humans sure are greedy.

0:24

Actually, quantum mechanics forbids this

Understood nothing but…Hotel was full..

The video gives me infinite confused.

This is my infiniteTh time wacthing this

Our brains cannot handle the concept of infinity, but what's more important than understanding eternity or a void without time, is The Eternal Now. What is now? That is the ultimate question.

"But over at Hilbert's infinite hotel, where there is never any vacancy and always room for more" — There actually is vacancy. An infinite number of vacancy, since all the numbers that are not powers of prime numbers are vacant (ex: 6) Great video btw!

Could’ve just said keep counting till there’s no more numbers that’s the end of infinity

This is just getting RIDICULOUS 😁😂😂

Im 12 and i understand this

This is legit what you see during the elevator game.

If I were the guest who was asked to move to another room by the hotel manager, I would just move to another Infinite Hotel nearby😂🤣

Room root -1 😎

Just book in another hotel

İf we have infinite room why we have problem? 🙂

Paradox?

a bad hotel wouldnt go there

Hi

Infinity hotel got out of rooms

Laughs in math.

Infinity stones.

How is an infinity bus structurally sound?

The elevator would need to be infinitely fast so that the people on the highest floors don't die of hunger on the way there.

First of all, the bus would never stop letting people off because there is an infinite amount of people. We would actually be stuck at watching the bus drive by us FOR-EV-ER!

it doesn’t matter if you move from n to n+1 or n+40 only 1 room is free that way isnt it? (1 room for some time until everyone moves to next room in the end 1 guyest wont have room 🤷♂️) p.s. or i dont understand

Oh the elevator break, i need walking to room number omega+1

infinite cannot get full

n how can the manager tell infinite people to move on to other room

So there is the counteble numbers of INFINITY rooms then there is no in INFINITY rooms |"({[SOLVED]})"|

THEN THE HOTEL IS OUT OF MONEY

Yeah, this is big brain time.

"Nieskończony hotel jest pełen"…

I to zdanie kończy całkowicie sens dalszych rozważań.

I will give $100,000 to anyone who counts how many times he says rooms and infinite

If we were to not ignore the fact that you have to move each time new guests come, the hotel would be infinitely empty.

When you have to be at the airport in 30 minutes but your on floor 34 Krillion

Doesn’t even make sense though how could an infinitely large hotel be full? If it was full it would be finite xD even if there were infinite guests it still doesn’t make sense

One factor he missed: considering the mind state of residence who are told to move to the next room after few billion shifts.

the shining intensifiesIf its infinity then why dont you just give your infinity custmers new rooms instead of moving people

Wait, so if there is an infinite number of bus with infinite passagers inside, that means that the customers will stay more on the corridors of the hotel more than in the rooms.

mathematicians when they have nothing to do

Why am I more scared by the number infinite then I am by horror movies

IM SORRY BUT I CAN'T UNDERSTAND IT.

Infinite hotel: we have infinite money

Inflation: allow me to introduce myself

But steel is heavier than feathers…

doesnt matter anyways cuz everyone is gonna leave since the manager keeps moving them

I don't get it. How can he lose money by not accepting the new bus loads of infinite people when he's already making an infinite amount of money from the infinite amount of people already staying at the hotel?

5:21 ‘never any vacancy’? You just said room 6 wouldn’t be filled if an infinite number of infinitely long buses with finite numbers of people checked into the hotel that night. Otherwise super interesting video! I watched this a year or two ago and wasn’t really listening, but watching it now, it makes much more sense!

It is a Galileo's paradox, Hilbert only represented it to his colloquium …

Boy that’s a whole it steps for those poor sods to take 🙁

Bookings.com wants to know your location.

Am i the only one who don't understand?

Thought experiments like this are just exercises in over complication

hmmm, I thought infinite means infinite thus, the infinite hotel could not possibly ever be full. Night manager can go back to sleep.

حرام كلشي مافتهمت بس غرف وفندق

but if its infinite then why not just do the way he did it before but instead of telling room himself just tell the person staying in the room to tell the next person to move down one

1, 2, 3, pi, 4, 5

just shut up

My brain explode now

how much do the cleaners get paid?

The first two examples used the Peanno definition of natural numbers.

I was surprised with the example using prime numbers, that was pretty cool!

I love studying math <3

or in room pi, where people expect free dessert. Haha

An infinite number of rooms is full.. Uhh, how?

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Infinititties